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Revision 2 1

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Physical basis of trace element partitioning: A review 3

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Shun-ichiro Karato 6

Yale University 7

Department of Geology and Geophysics 8

New Haven, CT 06520, USA 9

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submitted to American Mineralogist (Centennial Volume) 12

revised April 22, 2016 13

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key words: element partitioning, point defects, the Onuma diagram, strain energy model, 15

hard sphere model, noble gas 16

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Abstract 20

Experimental observations on the dissolution of elements in minerals and melts 21

and the partitioning between the two materials show that the concentration (or the 22

partition coefficient) of trace elements depends on the properties of elements as well as 23

those of relevant materials (minerals and melts) and the thermochemical conditions. 24

Previous models of element solubility in minerals contain a vague treatment of a role of 25

the stiffness of the element and have a difficulty in explaining some observations 26

including the solubility of the noble gases. A modified theory of element solubility in 27

minerals is presented where the role of elasticity of both matrix mineral and the element 28

is included using the continuum theory of point defects by Eshelby. This theory provides 29

a framework to explain a majority of observations and shows a better fit to the published 30

results on the effective elastic constants relevant to element partitioning. However, the 31

concept of “elasticity of the trace element” needs major modifications when the site 32

occupied by a trace element has large excess charge. The experimental data of the 33

solubility coefficients of noble gases in the melts show strong dependence on the atomic 34

size that invalidates the ‘zero-charge’ model for noble gas partitioning. A simple model 35

of element solubility in the melts is proposed based on the hard sphere model of complex 36

liquids that provides a plausible explanation for the difference in the dissolution behavior 37

between noble gases and other charged elements. Several applications of these models 38

are discussed including the nature of noble gas behavior in the deep/early Earth and the 39

water distribution in the lithosphere/asthenosphere system. 40

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INTRODUCTION 42

The distribution of elements in various materials on Earth has been used to infer 43

the chemical evolution of Earth including the history of partial melting and degassing that 44

has created the crust, atmosphere and oceans (e.g., (Allègre, 1982; Allègre et al., 45

1986/1987; Hofmann, 1997; Matsui et al., 1977)). The distribution of elements is 46

controlled largely by the difference in the excess free energy of a given element in co-47

exiting materials (e.g., (Blundy and Wood, 2003; Matsui et al., 1977; Nagasawa, 1966)), 48

although kinetic factors might also contribute if diffusion is slow (e.g., (Lee et al., 2007; 49

Van Orman et al., 2002)). When we assume chemical equilibrium to simplify the 50

discussion, then the element distribution is controlled by the differences in the excess free 51

energy of elements in coexisting materials such as minerals and melts. 52

The concentration of trace elements in minerals and melts changes with the 53

physical and chemical conditions as well as the properties of minerals (melts) and 54

elements. Consequently, understanding the controlling factors of concentration of 55

elements in minerals and melts will help us understand the physical and chemical 56

processes in Earth. This is an area where mineralogists (mineral physicists) can make an 57

important contribution to geochemistry. 58

Obviously, the most direct and crucial studies would be the experimental studies 59

on element partitioning but experimental studies of partitioning (solubility 1 ) are 60

challenging and the data set is incomplete particularly under the deep Earth conditions. In 61

some cases, there are large discrepancies among published results (e.g., a case of noble 62

gas partition coefficients in olivine and clinopyroxene: (Broadhurst et al., 1992; Hiyagon 63

1 I use the term “solubility” in a broad sense meaning the amount of an element in a material in the given thermo-chemical environment.

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and Ozima, 1986)). In case of Ar, for example, even the issue of either Ar behaves like 64

compatible or incompatible element upon partial melting (or solidification from the melt) 65

is controversial (e.g., (Broadhurst et al., 1992; Shcheka and Keppler, 2012; Watson et al., 66

2007)). Understanding the theoretical basis for dissolution of elements will help assess 67

the experimental observations. 68

In most of geochemical studies, we focus on the partitioning of trace elements 69

(elements with small concentration) because they are believed to behave as a passive 70

marker of physical/chemical processes (such as partial melting) without changing the 71

nature of the processes themselves. In these cases, the essence of theory of solubility of 72

trace elements in minerals is much the same as the theory of point defects in solids: both 73

point defects and trace elements are “impurities” in nearly perfect crystals. Therefore, the 74

results of a large amount of theoretical and experimental studies on point defects in solids 75

(for review, see e.g., (Eshelby, 1956; Flynn, 1972)) can be used to help understand the 76

physical mechanisms of element solubility (partitioning). In case of dissolution of trace 77

elements in the complex liquids (melts), somewhat different models will apply since the 78

structure and the thermodynamic properties of complex liquids are quite different from 79

those of solids (e.g., (Barrat and Hansen, 2003; Jing and Karato, 2011)). 80

In this paper, I will first review the basic observations on element partitioning, 81

summarize thermodynamics of element partitioning, and then discuss the physical models 82

of element solubility (uptake) including previously published models (Blundy and Wood, 83

1994, 2003; Carroll and Stolper, 1993; Guillot and Sarda, 2006; Nagasawa, 1966). In the 84

case of the solubility in solids, the previous models have a common limitation in 85

explaining why different elements and minerals have different partitioning, the most 86

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important goal of a theory. The limitation of the previous models becomes serious when 87

one considers the solubility (partitioning) of noble gas elements that have unusually 88

smaller “stiffness” than the host crystal. I will present a modified theory of element 89

partitioning to rectify this and finally discuss some implications. 90

In case of the liquids (melts), some theoretical models were proposed to explain 91

the solubility of noble gases (Carroll and Stolper, 1993; Guillot and Sarda, 2006). 92

However, the applicability of these models to other trace elements is unknown. I will 93

present a simple conceptual model of element dissolution in the melts based on the hard 94

sphere model of complex liquids (e.g., (Guillot and Sarda, 2006; Jing and Karato, 2011)) 95

and suggest that the dissolution mechanisms in the melts are different between neutral 96

elements (noble gases) and charged elements: noble gas elements go to the void space 97

while other charged trace elements replace an ion in the molecular cluster. 98

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EXPERIMENTAL OBSERVATIONS ON ELEMENT PARTITIONING AND 100

SOLUBILITY 101

102

The equilibrium distribution of an element between two materials can be 103

characterized by a partition coefficient that describes the ratio of concentration of a given 104

element between two materials. The concentration of an element can be defined in a few 105

different ways, and therefore there are several definitions of the partition coefficient (e.g., 106

(Blundy and Wood, 2003)). In most geochemical literatures, the concentration of an 107

element is measured by the weight fraction (as oxides in many cases) and the Nernst 108

partitioning coefficient is used that is defined by 109

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%DiY / X =

%CiY

%CiX (1) 111

112

where %CiY X( ) is the mass fraction of element i in phase Y (or X) (all the symbols used in 113

this paper are summarized in Table 1). Instead of the mass fraction, the molar fraction 114

may be used to define the molar partition coefficient, 115

116

DiY / X = Ci

Y

CiX (2) 117

118

where CiY X( ) is the molar fraction of the element i in phase Y (or X). An alternative 119

measure of element partitioning is the equilibrium constant, 120

121

KiY / X = ai

Y

aiX (3) 122

123

where aiY X( ) is the activity of element i in phase Y(X). Thermodynamically this is the 124

simplest definition because KiY / X contains only the thermodynamic properties of pure 125

end-member components. The molar partition coefficient, DiY / X , is identical to the 126

equilibrium constant when the activity of an element in a given material is the same as its 127

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molar concentration (i.e., ideal solution). I will make this assumption for simplicity and 128

review the models for DiY / X (or Ki

Y / X )2. 129

There is an extensive literature on element partitioning (for reviews, see (Blundy 130

and Wood, 2003; Jones, 1995; Wood and Blundy, 2004)). There are several features in 131

element partitioning (or solubility) that should be explained by a physical model: 132

(i) For a given pair of materials (say clinopyroxene and basaltic melt) at a given 133

physical/chemical condition, trace elements with different sizes and electric 134

charges have different partition coefficients (e.g., Fig. 1a, (Blundy and Dalton, 135

2000; Onuma et al., 1968)). 136

(ii) For a given element, partition coefficients depend strongly on minerals (and137

sometimes on melts). An important case is the contrast between Mg-perovskite 138

and Ca-perovskite (e.g., Fig. 1b, c, (Corgne et al., 2004; Hirose et al., 2004)) and 139

between diopside and olivine (e.g., (Witt-Eickschen and O'Neill, 2005)). 140

(iii) Even for the same pair of materials, partition coefficient of some elements (say141

hydrogen) depends strongly on thermo-chemical conditions such as temperature, 142

pressure and the fugacity of relevant species. A case is the hydrogen partitioning 143

between olivine and orthopyroxene (e.g., Fig. 1d, (Dai and Karato, 2009)). 144

(iv) The noble gas partition coefficient between olivine, diopside and the melt is145

nearly independent of the size of noble gas atom (e.g., Fig. 1e, (Brooker et al., 146

2003; Heber et al., 2007)) whereas the solubility of noble gas atom in bridgmanite 147

strongly depends on the atomic size (Fig. 1f, (Shcheka and Keppler, 2012)). 148

2 There are a few important cases where this assumption is not valid. In these cases, the role of fugacity of relevant species is important.

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(v) The solubility coefficient of noble gas in the melt is highly sensitive to the atomic 149

size of the noble gas (e.g., Fig. 1g, (Carroll and Stolper, 1993; Heber et al., 2007; 150

Shibata et al., 1998; Shibata et al., 1994)). 151

The issues listed as (i) through (iii) are all fundamental to the physics and 152

chemistry of element partitioning, but previous models to explain these features are 153

highly limited as I will discuss in the next section. For example, the influence of the 154

properties of trace element on the partitioning was not properly formulated in the 155

previous models. Also, the issue (iii), i.e., the sensitivity of the partition coefficient on 156

physical and chemical conditions has not been fully appreciated. Although this is a 157

consequence of a general physics and chemistry of element partitioning (see the next 158

section), the element partition coefficient is often considered to be a constant rather than 159

a property that depends on the physical/chemical conditions. Important cases are 160

hydrogen partitioning between olivine and orthopyroxene (Dai and Karato, 2009) and the 161

H/Ce ratios in basaltic magmas (Dixon et al., 2002) both of which have important 162

ramifications to the study of distribution of water (hydrogen). 163

Solubility and partitioning of the noble gases require special attention. 164

Understanding the behavior of noble gases is important because they provide important 165

clues to the evolution of Earth and other terrestrial planets (e.g., (Allègre et al., 1983; 166

Marty, 2012; Ozima, 1994)). Noble gas atoms have weak chemical bonding to other 167

atoms and hence the free energy change caused by the dissolution of noble gas atoms into 168

minerals and melts can be markedly different from those of other trace elements where 169

charged trace elements (e.g., H + , La3+ ,Sm3+ ,U 4+ ) replace other cations (e.g., 170

Mg2+ , Ca2+ , Al3+ , Si4+ ) in the host minerals or melts. 171

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172

THEORETICAL MODELS FOR ELEMENT SOLUBILITY (PARTITIONING) 173

174 Chemical reactions and thermodynamics 175

176 Physics of element partitioning can be viewed in a few different ways. When one 177

considers element partitioning between melt (liquid) and mineral (solid), one could 178

imagine a model where a trace element i and host element h are exchanged between a 179

liquid and a solid (Fig. 2a)3, viz., 180

181

S y( ) + L i( ) ⇔ S i( ) + L y( ) (4) 182

183

where S y( ) is a solid (mineral) containing cation y, and L i( ) is a liquid (melt)184

containing element i etc. The free energy change associated with the reaction (4) can be 185

calculated by dividing the reaction into two separate reactions (Fig. 2a), namely, 186

187

S y( ) → L y( ) (5a) 188

and 189

L i( ) → S i( ) . (5b) 190

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3 In reality, there are several cases where the exchange of multiple elements is involved in the dissolution of some elements (“coupled substitution”, e.g., Al3+ + H + ⇔ Si4+ ). To simplify the discussion, I will focus on simple cases (without coupled substitution), and will discuss the issues of coupled substitution only briefly in relation to hydrogen and noble gas partitioning.

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The reaction (5a) is melting of the host mineral, and the free energy change 192

associated with this reaction is the free energy change upon melting ( ΔG fusiony in the 193

notation by (Blundy and Wood, 1994)). The reaction (5b), L i( ) → S i( ), is a reaction to 194

bring a trace element from the liquid to the solid (the free energy change is ΔGexchangey−i in 195

(Blundy and Wood, 1994)). Blundy and Wood (1994) argued that the latter is dominated 196

by the strain energy in solid and developed a model using the theory by (Brice, 1975). In 197

this treatment, the role of liquid (melt) is obscured because the free energy change in the 198

liquid (melt) is included only implicitly in ΔGexchangey−i . In fact, I will show that the role of199

liquid is important in the case of noble gas where the dissolution of noble gas in the melt 200

has non-negligible excess free energy. 201

The physical nature of the change in the free energy associated with element 202

partitioning between a solid and a liquid can be understood more clearly by considering 203

the chemical reactions of both a solid (mineral) and a liquid (melt) with an “environment 204

(a reservoir)” that is a fluid phase (Fig. 2b). In this approach, I consider element 205

dissolution in a solid and a liquid separately, and by taking the ratio of the concentration 206

of a given element in a solid and a liquid, I will calculate the partition coefficient. The 207

chemical equilibrium of material X (either a solid or a liquid) with a reservoir A with 208

respect to the exchange of elements i and y can be written as 209

210 A i; y − 1( ) + X i − 1; y( ) = A i − 1; y( ) + X i; y − 1( ) (6) 211

212

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where A i; y − 1( ) is a reservoir containing the element i and y-1, X i; y − 1( ) is a phase X213

that contains an element i as an impurity and the host cation y-1. Rewriting equation (6), 214

one gets, 215

216

%A i; y( ) = %X i; y( ) (7) 217

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where %A i; y( ) = A i; y − 1( ) − A i − 1; y( ) and %X i; y( ) = X i; y − 1( ) − X i − 1; y( ). The219

chemical equilibrium of reaction (7) demands 220

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μ %A i;y( ) = μ %X i;y( ) (8) 222

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where μ %A i;y( ) is the chemical potential of the reservoir (A) containing element i and y and224

μ %X i;y( ) is the chemical potential of phase X (solid or liquid) containing element i and y. I225

assume that the reservoir is large, and therefore the properties of the reservoir are 226

insensitive to the amount of host element, i.e., μ %A i;y( ) ≈ μ %A i( ) . Then227

228

μ %A i( ) = μ %A i( )o + RT log f %A i( )

Po(9) 229

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where f %A i( ) is the fugacity of element i in the reservoir %A i( ) , Po is the reference pressure231

and μ %A i( )o is the chemical potential of the reservoir at the reference pressure (and232

temperature). 233

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As to the chemical potential of X (a solid or a liquid) containing some trace 234

element, μ %X i;y( ) , I will assume that the amount of trace element is small and hence the235

change in the concentration of the host cation (say from y to y-1) does not change the 236

chemical potential of the solid (or the liquid), μ %X i;y( ) ≈ μ %X i( ) . This assumption also leads237

to an ideal solution model where the chemical potential of a solid (or a liquid) containing 238

the trace element is given by, 239

240

μ %X i( ) = μ %X i( )0 + RT logCi

X (10) 241

242

where μ %X i( )0 is the change in chemical potential of phase X by replacing the host cation 243

(y) with a trace element (i) (i.e., the formation free energy of a “defect”), CiX is the 244

(molar) concentration of element i in a phase X, and RT has their usual meaning4. 245

In writing the chemical equilibrium between the reservoir (a fluid phase) and a 246

solid (or a liquid), it is necessary to know how many molecules of the fluid phase are 247

involved in the given reaction. For instance, when Ar is dissolved in a mineral, one may 248

write 249

250

Ar + X = X Ar( ) (11) 251

252

where one mole of Ar reacts to form a mineral containing a certain amount of Ar. The 253

situation is different in case of the dissolution of hydrogen H in a mineral. In this case, 254

4 In a more realistic case, where trace elements interact each other, one needs to make a correction to the relation (8) by introducing the activity coefficient.

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the above formula must be modified because hydrogen can react with other species such 255

as oxygen O to form a “compound” such as OH or 2H( )M× (two protons trapped at the256

M-site vacancy). In this case, we can write the chemical reaction equation as257

258

α ⋅ H2O + X = X H( ) (12) 259

260

where α is a constant that depends on the nature of H-bearing chemical species 261

contained in phase X such as OH or 2H( )M× (two protons trapped at M-site). In case of262

OH, α =½, whereas in case of 2H( )M× , α =1 (e.g., (Karato, 2008)).263

Therefore the concentration of element i in mineral X that co-exists with a 264

reservoir for element i (a fluid phase A) is given by 265

266

CiX,A = exp α i

Xμ %A i( )0

RT( ) ⋅ f %A i( ) P,T( )Po

⎡⎣⎢

⎤⎦⎥

α iX

⋅exp − μ %X i( )0

RT( ) . (13) 267

268

CiX,A is the solubility of element i in mineral X if the fluid phase A is the end-member 269

phase (e.g., if the fluid phase is water, then equation (13) will be the solubility of 270

hydrogen (i) in mineral X). The same formula applies to another material (mineral or 271

melt), Y. The ratio of concentration of a species i between two phases (Y and X) can then 272

be given by 273

274

CiY ,A

CiX ,A = Di

Y / X = exp α iX −α i

Y( )μ %A i( )o

RT⎡⎣

⎤⎦ ⋅ f %A i( )

α iX −α i

YP,T( ) ⋅exp − μ %Y i( )

0 −μ %X i( )0

RT( ) (14)275

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276

where α iY X( ) are the coefficients in chemical reaction such as (11), Di

Y / X is the molar 277

partition coefficient of an element i between the phase Y and X, that is equivalent to the 278

molar partition coefficient if the ideal solution model works. 279

A simple but important and general conclusion from the relations (13) and (14) is 280

that the partition coefficient is generally a function of thermodynamic conditions 281

including pressure, temperature and the fugacity of relevant components. In other words, 282

the tendency of an element to prefer one phase over another phases depends on the 283

thermodynamic conditions. Consequently, experimental data on element partitioning 284

obtained under some limited conditions should not be applied to largely different 285

conditions without proper corrections. For instance, the partition coefficient of hydrogen 286

between olivine and orthopyroxene changes with pressure, temperature and water 287

fugacity by a large amount, say a factor of 10 or more ((Dai and Karato, 2009), see also 288

(Sakurai et al., 2014)). 289

A case of trace element partitioning can be treated easily if one considers the 290

partitioning of elements with similar chemical properties (e.g., partitioning of rare Earth 291

elements). In such a case, one can assume α iX = α i

Y , and the term containing the 292

properties of the reservoir can be eliminated, and equation (14) becomes 293

294

CiY ,A

CiX ,A = Ci

Y

CiX = exp − μ %Y i( )

0 −μ %X i( )0

RT( ). (15) 295

296

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Consequently, in this case the only task is to evaluate how μ %Y i( )0 − μ %X i( )

0 depends on the 297

properties of the element and the matrix. 298

In the literatures, the element partitioning between the melts and minerals is 299

discussed (e.g., (Blundy and Wood, 2003; Matsui et al., 1977; Nagasawa, 1966; Onuma 300

et al., 1968; Wood and Blundy, 2004)). In these cases, although the influence of melt 301

composition is studied (e.g., (Blundy and Dalton, 2000; O'Neill and Eggins, 2002; 302

Schmidt et al., 2006)), it is often assumed that the excess energy for the melt is 303

independent of the trace element ( μ %Y i( )0 is independent of trace element, i), and the304

discussion is focused on the excess energy in minerals ( μ %X i( )0 ). Noble gases are305

exceptions: their solubility in melts is small and highly sensitive to the atomic size of the 306

noble gas, the solubility (coefficient) varies more than a factor of ~100 among different 307

species (e.g., (Carroll and Stolper, 1993; Heber et al., 2007; Shibata et al., 1998)). The 308

physical reasons for different behavior will be discussed in a later section based on the 309

hard sphere model of silicate melts. 310

When the role of melts is minor, then the main question is what determines the 311

different solubility of different elements in different minerals (what controls the degree of 312

“incompatibility”)? One may ask two different questions: (1) why different elements 313

have different solubility in a given mineral?, and (2) why different minerals have 314

different solubility for a given element? Such questions were addressed by a pioneer of 315

geochemistry, Goldschmidt, who also classified elements into several categories based on 316

the affinity to various materials (Goldschmidt, 1937). Goldschmidt pointed out that the 317

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size of ions and the crystal structure of minerals (as well as the charge of the ion) are the 318

key to determine the partition coefficients5. 319

The breakthrough on this topic was made by Onuma, Matsui and their colleagues 320

(Matsui et al., 1977; Nagasawa, 1966; Onuma et al., 1968) who clearly showed that the 321

partition coefficients of elements between minerals and melts depend on the size (ionic 322

radius) of that element relative to the size of the site in the mineral as envisaged by 323

Goldschmidt. A diagram showing the partition coefficients as a function of ionic radius is 324

called the Onuma diagram. 325

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Outline of the models for the excess free energy 327

The excess free energy associated with the dissolution of a trace element may be 328

calculated from theoretical models incorporating the atomistic details (e.g., (Allan et al., 329

2001; Purton et al., 1996; Purton et al., 2000)), but the use of simpler theoretical models 330

will make the basic physics clearer. Therefore, I will focus on the theoretical models from 331

which some essence of element solubility can be understood. 332

Two types of models will be considered. In case of a solid (a mineral), the change 333

in free energy caused by the dissolution of trace element is dominated by the change in 334

enthalpy, i.e., the change in internal energy and volume. A trace element in a crystal can 335

be considered as a point defect, and therefore the change in internal energy and volume 336

associated with trace element dissolution may be formulated following the models of 337

point defects (e.g., (Eshelby, 1954, 1956; Flynn, 1972; Mott and Littleton, 1938)). 338

5 “One of the most important principles for the distribution of the elements is the grading according to their size, especially as compared with the lattice spacings or interatomic distances of rock-forming minerals” (from Goldschmidt (1937)).

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The excess free energy caused by a point defect can be calculated by the response 339

of a crystal to the introduction of a point defect. A point defect exerts some force to the 340

surrounding crystal. This force could be separated into two components, the force caused 341

by the “size” mismatch and the force caused by the excess charge. The first force causes 342

uniform displacement of cations and anions, while the second one causes opposite 343

displacement between cations and anions, i.e., the dielectric polarization. The excess free 344

energy associated with the first can be expressed as the strain energy, while the latter as 345

the electrostatic energy. In some cases, these two effects are related and I will come back 346

to that point when I discuss the dissolution of noble gases. Although this is a gross 347

simplification of the actual processes of formation of point defects (or the dissolution of 348

trace elements), such an approach provides a good estimate of some properties of point 349

defects in olivine and other minerals (e.g., (Karato, 1977, 1981; Lasaga, 1980)). 350

Dissolution of a trace element in complex liquids such as silicate melts should be 351

treated in a different way because thermodynamic properties of complex liquids are 352

markedly different from those of solids (minerals). The differences in thermodynamic 353

properties include small (and a narrow range of) bulk moduli that are unrelated to the 354

bulk moduli of corresponding solids and the positive pressure dependence of Grüneisen 355

parameter. Jing and Karato (2011) showed that most of these observations can be 356

explained by a hard-sphere model in which the main contribution to the free energy is the 357

configurational entropy rather than the enthalpy. In this model, a silicate melt is 358

considered to be a mixture of hard spheres and free space, and the motion of hard spheres 359

in the free space contributes to the configurational entropy. A trace element could go 360

either into the free space (void space) or into the hard spheres by replacing the pre-361

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existing ions in them. I will discuss these two cases based on the hard sphere model of a 362

complex liquid in a later section. 363

364

Continuum models of excess energy for a solid 365

When an atom (or an ion) in a crystal is replaced with another one (I will call it as 366

a trace element), the free energy of the crystal will change. The change in chemical 367

potential (the Gibbs free energy per mole) is generally expressed as 368

369

Δμ = Δu + PΔυ − T Δs = Δh − T Δs (16) 370

371

where Δμ is a change in the chemical potential, Δu is a change in the internal energy, 372

Δυ is a change in the volume, and Δs is a change in the entropy ( Δh is a change in 373

enthalpy). In solids, the entropy change in this equation corresponds to a change in the 374

vibrational entropy (Flynn, 1972). In general a change in the vibrational entropy caused 375

by a point defect is a fraction of R (measured by J/K/mol) (e.g., (Maradudin et al., 1971)), 376

and the influence of this term on element partitioning is small ( exp ΔsR( ) ≈ O 1( )). The377

pressure effect PΔυ is important when partition coefficient under a broad pressure range 378

is investigated. However, the emphasis in this paper is to provide a good explanation for 379

the behavior of partition coefficient for different elements or for different materials 380

(under the limited pressure (and temperature) conditions), so I will focus on Δu . 381

In contrast, in silicate melts, the change in configurational entropy can be large 382

when a trace element atom occupies the “free space” (or the void space). This is likely 383

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the case for the dissolution of noble gases in melts (e.g., (Carroll and Stolper, 1993; 384

Guillot and Sarda, 2006)). In these cases, the entropy term cannot be ignored. 385

The trace element that replaces an ion in the host crystal has in general a different 386

size and charge than those of the ion in the matrix that will be replaced with the trace 387

element. Therefore the dissolution of a trace element creates excess elastic strain and 388

dielectric polarization of a crystal (in addition to the change in the strain energy of the 389

trace element itself). In the continuum approximation, the excess energy may be written 390

as 391

392

μ %Y i( )0 = μ %Y i( )

elastic + μ %Y i( )dielectric (17) 393

394

where μ %Y i( )elastic is the excess elastic strain energy and μ %Y i( )

dielectric is the excess dielectric 395

polarization energy. 396

Under these approximations, the partition coefficient can be written as 397

398

DiY / X = Di

Y / X,elastic ⋅ DiY / X,dielectric

≈ exp − Δuelastic

RT( ) ⋅exp − Δudielectric

RT( ) (18) 399

400

where DiY / X,elastic ≈ exp − Δuelastic

RT( ) , DiY / X,dielectric ≈ exp − Δudielectric

RT( ) and the quantities are401

for one mole, and the symbols i, Y/X are removed in the second line for simplicity. 402

403

Strain energy models 404

20

Now let us focus on the contribution from the elastic strain, Δuelastic . The 405

replacement of an ion with a trace element with a different size ( ro : the radius of the site 406

at which a trace element is placed, r1 : the radius of the trace element) results in the 407

excess elastic strain energy. The strain energy is determined by the magnitude of strain 408

caused by this replacement and the elastic properties of both the matrix crystal and of the 409

trace element. Therefore the key here is to calculate (i) the magnitude of strain and (ii) 410

the strain energy associated with this process. 411

(Nagasawa, 1966) was the first to discuss the nature of trace element partitioning 412

based on the strain energy model. He used a theory by (Eshelby, 1954) and calculated the 413

strain energy associated with the dissolution of a trace element assuming that the elastic 414

properties of the matrix crystal are the same as those of the trace element. This 415

assumption is valid only when the bulk moduli (only bulk modulus matters inside the 416

inclusion according to the theory of (Eshelby, 1954)) of the matrix and the trace element 417

are the same. In a more general case, the influence of different elastic properties of the 418

matrix and the trace element needs to be included. Also, Nagasawa (1966) ignored the 419

influence of the image force (Eshelby, 1954, 1956) causing small differences in the 420

formula for the effective elastic constant (see also Table 2). 421

Blundy and Wood (e.g., (Blundy and Wood, 1994, 2003; Wood and Blundy, 422

1997, 2001, 2004)) used a model by (Brice, 1975) to interpret a large number of 423

experimental data including the partitioning of noble gases. Their model is similar to that 424

by (Nagasawa, 1966), but the model by (Brice, 1975) contains a few physically unsound 425

assumptions. For instance, Brice assumes that when a trace element with the radius r1 is 426

inserted to a site with the radius ro , then the radius of the site changes to r1 . This is 427

21

correct only when the trace element is infinitely stiff. In a more general case where the 428

trace element has a finite bulk modulus, the displacement is not only controlled by the 429

size difference but also by the difference in the elastic constants. This leads to a large 430

systematic error when the trace element is much softer than the matrix, a case for the 431

noble gas. Furthermore, Brice used an incorrect expression for the strain both inside and 432

outside of the inclusion. The strain field in the matrix surrounding a spherical inclusion is 433

shear strain and the strain inside of the inclusion is compressional strain (Eshelby, 1956), 434

but Brice used a Young’s modulus and did not pay attention to the difference in the strain 435

field inside and outside of an inclusion. Despite these differences, these two models give 436

similar equations (Table 2), and both of them explain some of the experimental 437

observations (e.g., the Onuma diagram for some elements). 438

In short, these previous models have common limitations in ignoring the 439

difference in the elastic properties between the matrix and the trace element (impurity). 440

An important case is the partitioning of noble gas elements where the trace element 441

(impurity) has much smaller bulk modulus than the matrix. In such a case, the strain 442

would be small ε = %rro

− 1 1 much less than the Brice model would predict ε = r1r0

− 1.443

The appropriate treatment of the role of the size and stiffness of the trace element 444

is a key step in understanding how the properties of the trace elements and of the matrix 445

affect element partitioning. As will be shown later, the stiffness of the trace element has a 446

strong influence on the magnitude of lattice strain and therefore it is one of the key 447

parameters controlling the strain energy. To rectify the limitations of these previous 448

models, I have made modifications to the continuum model of trace element solubility by 449

introducing the following three points: (i) the proper boundary conditions at the boundary 450

22

between the inserted (trace) element and the surrounding matrix (i.e., the continuity of the 451

displacement and the normal stress) are included in solving the equation for the 452

equilibrium conditions (this was included in (Nagasawa, 1966) but not in (Brice, 1975)), 453

(ii) the strain energy of the element itself is included in addition to the strain energy of454

the host crystal using the different elastic moduli (in previous models, strain energy in the 455

trace element was calculated assuming the same elastic constant as the matrix) and (iii) 456

both volumetric and shear strain are considered (this was correctly included in 457

(Nagasawa, 1966) but Brice used an incorrect relationship for the strain). 458

An analysis including these points shows that the displacement of the boundary 459

caused by the replacement of an atom (ion) with the radius ro with that of a trace element 460

with the radius r1 depends not only on the relative size but also on the elastic constants of 461

the trace element and of the matrix as (Fig. 3; see also Appendix 1) 462

463

ε = %rro

− 1 = K1K1+ 4

3 Go

r1ro

− 1( ) (19) 464

465

where %r is the final (equilibrium) size of the site now occupied by a trace element, Go is 466

the shear modulus of the matrix and K1 is the “bulk modulus” of the trace element6.467

Equation (19) means that if the trace element is very stiff compared to the shear modulus 468

of the matrix ( K1 Go ), then %r ≈ r1 , whereas for a weak trace element ( K1 Go ), 469

(and ε ≈0) (Fig. 4). This concept plays a key role in explaining the solubility 470

(partitioning) of noble gas elements. Corresponding to this displacement, both the 471

6 Physical meaning of the bulk modulus of a trace element in a lattice site can be complicated and will be discussed in the later part of this paper.

23

element itself and the crystal will undergo elastic deformation leading to an increase in 472

the strain energy (per one trace element) that is given by (Appendix 1), 473

474

Δuelastic = 6πro3 K1

2

K1+ 43Go

r1ro

− 1( )21+ K1

K1+ 43Go

r1ro

− 1( )⎡⎣⎢

⎤⎦⎥. (20) 475

476

This equation contains the bulk modulus of a trace element ( K1 ) and the shear 477

modulus of the matrix crystal ( Go ). This corresponds to the fact that the strain inside of a 478

spherical inclusion is homogeneous compression while the strain outside of an inclusion 479

is shear strain (Eshelby, 1951, 1954, 1956). The effective elastic constant (the EEC, or 480

the lattice strain parameter) corresponding to the Young’s modulus in the Brice model 481

would be 3K12

K1+ 43 G0

that is related to the stiffness of the element as EEC ≈ 3K1 for 482

K1 Go , while EEC ≈ 94

K12

Go for K1 Go. Therefore the influence of elasticity of trace 483

elements is large when the elastic constant of the trace element is much different from 484

that of the matrix minerals. These predictions of the model have important bearing on the 485

interpretations of experimental observations (see Discussion). 486

487

Influence of excess charge: dielectric polarization energy and influence on strain 488

When a trace element goes to a site that is usually occupied by an ion with a 489

different electrostatic charge, then there will be an excess charge, either positive or 490

negative, relative to the perfect crystal at the site that the trace element occupies. The 491

excess charge exerts electrostatic force to the surrounding ions. Due to this force, cations 492

and anions will move to the opposite directions causing dielectric polarization. The 493

24

dielectric polarization energy caused by an excess electrostatic charge, ΔZ ⋅e, is given by 494

(e.g., (Flynn, 1972)), 495

496

Δudielectric = ΔZ( )2e2

2 %rκ (21) 497

498

where κ is the static dielectric constant and %r is the size of the defect. This energy 499

decreases with the size of the trace element, and hence the solubility of the element 500

increases with the size of the defect. However, the influence of the atomic size is weak 501

compared to that in the elastic strain energy (see equation (20) where r1r0

− 1( )2 term502

provides strong influence of the size of atoms (ions)). Its effect is to change the values of 503

partition coefficient by a similar amount for all the trace elements. Systematic differences 504

in the partition coefficients among different minerals (e.g., Mg-perovskite versus Ca-505

perovskite) might be due to the difference in the static dielectric constant, κ , between 506

these minerals (see a later section). 507

The static dielectric constant is the sum of the contributions from electronic, ionic 508

and dipolar effects and the dielectric constant varies among different minerals (e.g., 509

(Kittel, 1986)). In general, an ion with a large radius has a large electronic polarizability 510

that has an important contribution to the static dielectric constants. Ca2+ has substantially 511

higher electronic polarizability and hence Ca-bearing minerals tend to have a large 512

dielectric constant (e.g., (Shannon, 1993)). 513

Excess charge has another effect. A large part of the atomic displacement caused 514

by the excess charge is the anti-symmetric movement of cations and anions, i.e., 515

25

dielectric polarization. However, near the vicinity of the excess charge, displacement of 516

ions is large and can contribute to the elastic strain, ε = %rr0

− 1. Let us consider a case517

where Mg2+ at the M-site is replaced with Ar (i.e., A ′′rM (Ar at the M-site with two 518

negative effective charge)). In such a case, the effective negative charge (the negative 519

charge relative to the perfect lattice) is present at the M-site that exerts a large force to the 520

neighboring oxygen ions to cause their displacement away from the defect ( A ′′rM ). The 521

similar effect was observed by (Spalt et al., 1973) for a vacancy in KBr. Consequently, 522

one expects a larger elastic strain than expected from ε = %rr0

− 1 = K1K1+ 4

3Go

r1r0

− 1( ), leading to523

larger strain energy. I will come back to this issue when I discuss the partitioning of 524

noble gases (see also Appendix 2). 525

Finally, excess charge has another effect: the effect caused by the charge balance. 526

This is a chemical effect in the sense that in order to deal with the charge balance one 527

must consider the interaction with other charged species. This issue will be discussed 528

when I discuss the partitioning (dissolution) of noble gas elements and hydrogen (water). 529

530

Trace element dissolution in the melts 531

In the literature where the element partitioning between minerals and melts is 532

discussed, it is often assumed that the sensitivity of element partitioning on the atomic 533

(ionic) size of element is caused by the sensitivity of the solubility in minerals to atomic 534

(or ionic) size of elements, and that the element dissolution in melts is associated with 535

small excess energy and is insensitive to the size of elements (e.g., (Blundy and Wood, 536

2003)). This is the case for most trace elements. 537

26

Noble gases do not follow this: solubility is low and sensitive to the atomic size 538

(Carroll and Stolper, 1993; Guillot and Sarda, 2006; Guillot and Sator, 2012; Heber et al., 539

2007; Shibata et al., 1998; Shibata et al., 1994). There is no clear evidence of the 540

presence of a peak in the solubility coefficient when plotted against the size of noble gas 541

atom. Therefore these observations suggest that noble gas atoms do not occupy well-542

defined sites. Carrol and Stolper (1993) explained this observation by a model in which 543

noble gas atoms occupy the void space. Similarly, Guillot and his colleagues used a “hard 544

sphere model” in which they assumed that noble gas atoms occupy the void space among 545

the hard spheres (Guillot and Sarda, 2006; Guillot and Sator, 2012). The hard sphere 546

model also explains the correlation between the composition and the solubility coefficient 547

of noble gases: the noble gas solubility coefficient is higher in a melt with higher silica 548

content (Shibata et al., 1998). Such a trend is often explained by the concept of NBO 549

(non-bridging oxygen; (Mysen, 1983)), but this can also be explained by a hard sphere 550

model because the degree of net-working increases with the increase of the silica content 551

that leads to a higher void space (Guillot and Sarda, 2006; Guillot and Sator, 2012). 552

Given a marked dependence of noble gas solubility coefficient in the melt on their 553

atomic size but a commonly made assumption of independence of other trace element 554

dissolution on their ionic size, one may wonder why the dissolution behavior of these two 555

types of elements in the melts is so different. In order to understand what controls the 556

mechanisms of dissolution of elements in the melt, let us consider a hard sphere model of 557

silicate melts (Fig. 5). Unlike minerals, complex liquids such as silicate melts can be 558

considered as a mixture of clusters of atoms (hard spheres) that are randomly distributed 559

leaving void space among them. Since these clusters are separated by the void space, 560

27

their direct mutual interaction is weak except that any cluster cannot move into the space 561

occupied by other clusters (“excluded volume”). These clusters move nearly freely in the 562

limited space (space unoccupied by other clusters) and hence the internal energy of a 563

cluster does not change much with the type of liquid where it is located. Free motion of 564

clusters in the limited space contributes to the configurational entropy, Sconfig , and 565

−T ⋅Sconfig makes the dominant contribution to the free energy of a complex liquid. The566

hard sphere model is one of these models and provides a systematic explanation of a 567

large number of observations on the equation of state of melts (Jing and Karato, 2011). 568

Fig. 5a shows a case where a trace element (i) replaces a host element (y) in a 569

molecular cluster in the liquid. This is a case where the element i has modest electric 570

charge similar to the host ion h. The initial energy of the whole system is uinitial = uiA + uy

L571

( ui.yL,A : energy of a cluster in the liquid (L) or in the reservoir (A) containing the element i 572

or h), and the final energy is u final = uyA + ui

L . Therefore u final − uinitial = uyA − ui

A( ) − uyL − ui

L( ).573

In a hard sphere model, clusters (hard spheres) do not interact each other energetically. 574

Therefore the energy difference such as uyL,A − ui

L,A is the energy difference in the clusters 575

and uyL − ui

L ≈ uyA − ui

A , i.e., u final ≈ uinitial7. Since both elements i and h occupy the cluster,576

there is little change in the excluded volume and hence little change in the configurational 577

entropy and μ final ≈ μinitial . Consequently the solubility of these elements is high and 578

nearly independent of their size. 579

7 For a solid, ui,hL ≠ ui,h

A because of the strong interaction among the clusters, and hence uinitial

S ≠ u finalS .

28

For a noble gas element that has neutral charge, there will be a large excess 580

electrostatic energy if it replaces an ion in a cluster. Also the noble gas in the 581

environment (“A”) is not in the cluster, and a cation will not be dissolved in the noble 582

gas. Consequently, the noble gas dissolution does not occur as an exchange of a noble gas 583

atom and the cation in the liquid. Therefore the second mechanism (occupying the void 584

space) will be preferred (Fig. 5b). In this case, the excess energy strongly depends on the 585

size of the noble gas atom that determines the decrease in the void space (free volume). 586

587

DISCUSSION 588

589 Comparison with the previous strain energy models on element solubility in minerals 590

Elastic strain energy associated with the replacement of an ion in a mineral with a 591

trace element is an important factor controlling the solubility of the trace element in a 592

mineral (e.g., (Blundy and Wood, 2003)). In this section, I compare various strain energy 593

models with the experimental observations. Table 2 compares three models of the strain 594

energy associated with the dissolution of an element in a mineral, and Fig. 6 shows a 595

graph of normalized solubility (~partition coefficient if the element solubility in the melt 596

is independent of the size of the element) against r1ro

. All models show a peak in the 597

solubility at the ionic radius corresponding to the radius of the site of the host crystal 598

( r1ro

=1) (see equation (19)). The curvature of the curves is determined by the effective 599

elastic constant EEC( )obs relevant to element substitution that can be defined as 600

601

Δuelastic = 2πro3 EEC( )obs

r1ro

− 1( )21+ ξ r1

ro− 1( )⎡

⎣⎤⎦

(22)602

29

603

where Δuelastic is the strain energy and ξ = K1K1+ 4

3Go for my model and ξ = 2

3 for the Brice 604

model8. Although all models show similar curves, the curvature, i.e., EEC( )obs , has605

different expressions (see Table 2), and it is EEC( )obs that distinguishes different models. 606

EEC( )obs was calculated by Blundy and Wood for various sites in various minerals. For607

each combination of the site and the mineral, experimental data on partitioning for 608

various elements were used and from the shape of the curve of the Onuma diagram, they 609

calculated EEC( )obs (see Appendix 3). In the following, I will use the values of EEC( )obs 610

and compare them with the predictions from various models to evaluate the validity of 611

the models. 612

The simplest model for EEC( )calc would be the Brice model where all the relevant 613

elastic constants are those for the matrix. In this case, 614

615

EEC( )calcBrice = 3KoGo

Ko +Go /3 ≈ 1.5Ko = 0.225 Zo

ro +roxy( )3 (23) 616

617

where Ko ( EEC ) is in GPa, r in nm, and Zo is the valence of the ion at the site (+2 for 618

Mg2+ ), ro is the ionic radius of the site in the matrix that is replaced with the trace 619

element, roxy is the ionic radius of oxygen (0.138 nm). The results are compared with 620

EEC( )obs in Fig. 7a. The results show very poor fit indicating that the properties of a621

trace element other than its size play an important role in controlling the effective elastic 622

8 The difference in ξ between these two models is small and does not affect the calculated values of EEC( )obs substantially.

30

constant, EEC . I evaluate the goodness of the model by a parameter χ 2 (reduced chi-623

square or variance)9 and χ 2 =106 for this model. 624

Blundy and Wood (1994) noted a good correlation between EEC( )obs and Z1 625

(charge of the trace element) in plagioclase and diopside ( EEC( )obs ∝ Z1)). The EEC of 626

the M2 site of clinopyroxene systematically changes with the charge of the trace elements 627

in such a way that EEC( )M2+1 < EEC( )M2

+2 < EEC( )M2+3 where EEC( )M 2

+n is the effective elastic 628

modulus of the M2 site for trace elements with a charge +n (Blundy and Dalton, 2000). 629

Similarly, (Hill et al., 2011) found EEC( )M2+3

EEC( )M1+4 for clinopyroxene although the 630

polyhedron bulk moduli for the M1 and M2 sites are similar (Levien and Prewitt, 1981). 631

The EEC for a given mineral and a given site varies as much as a factor of ~100 among 632

different trace elements. This challenges the theory because none of the previous theories 633

(Brice, 1975; Nagasawa, 1966) includes the properties of a trace element other than its 634

size. 635

To account for the strong influence of the electrostatic charge of the trace element 636

Z1, Blundy and Wood proposed the following relationship, 637

638

EEC( )calcBW = 1.125 ⋅ Z1 / ro + roxy( )3 (24) 639

640

9 χ 2 (reduced chi-square or normalized variance) is defined as χ 2 = 1N

yji −x j( )2

i∑

yji −yj( )2

i∑j

∑ where j

specifies a combination of a mineral, the site and the charge of the trace element (j=1---N), and i specifies the individual data of EEC (i=1--- M j ), yj

i is the inferred value of EEC for a given i and j from the experimental data, x j is the model prediction for j and yj is the

mean value of yji . For the perfect model, χ 2 =0.

31

and called this as “site-elasticity” ( EEC( )calcBW in GPa, r in nm) (Blundy and Wood, 1994). 641

A comparison of this model with EEC( )obs is shown in Fig. 7b. This model better fits 642

EEC( )obs ( χ 2 =40) but shows a systematic deviation from EEC( )obs for the large values643

of EEC. Apart from the use of a dimensionally incorrect formula for an elastic constant10, 644

the theoretical basis of combining the property of the trace element ( Z1) and the property 645

of the matrix ( ro ) in the Blundy-Wood model is unclear. 646

In contrast to the previous models by (Nagasawa, 1966) and (Blundy and Wood, 647

1994), my model includes the influence of different elastic properties of the matrix and 648

the trace element based on the Eshelby theory of a point defect in an elastic material. This 649

model shows that the effective elastic constant (EEC) is ECC( )calcKarato = 3K1

2

K1+ 43Go

.650

According to this model, it is the bulk modulus of the trace element ( K1) and the 651

shear modulus of the matrix ( Go ) that determine the EEC. The EEC is not the Young’s 652

modulus of the material as incorrectly assumed by Brice (1975). For K1 , I use a 653

relationship K1 = 0.15 ⋅ Z1 / r1 + roxy( )4 11 corrected from (Hazen and Finger, 1979) and654

calculated EEC( ) calcKarato= 3K1

2

K1+ 43G o

. In other words, I assume that K1 is determined by the 655

bonding between the trace element and the surrounding oxygen ions. One problem with 656

this approach is that because EEC( )obs is calculated for each site (each ro ) for a range of 657

10 The relation EEC( )calcBW = 1.125 ⋅ Z1 / ro + roxy( )3 is derived from Hazen and Finger (1979)

model, K = 0.75 ⋅ Z / r + roxy( )3 (K in GPa, Z: charge of cation, r: radius of cation (nm)) butthis equation is dimensionally incorrect (Karato, 2008). A dimensionally correct equation is K = 0.15 ⋅ Z / r + roxy( )4 , but these two equations predict similar elastic constants.11 One could use a relation similar to Blundy-Wood’s model, i.e., the use of ro instead of

r1, K1 = 0.15 ⋅ Z1 / ro + roxy( )4 . The results are similar (not shown).

32

r1 one must use some average of r1. I used a simple arithmetic average. The calculated 658

EEC( )calcKarato are compared with EEC( )obs in Fig. 7c (for the details see Appendix 3). My 659

model, EEC( )calcKarato , shows a better fit to EEC( )obs for large values of EEC( ) , and the 660

variance is substantially reduced ( χ 2 =18 for this model). However, the use of the “bulk 661

modulus” to represent the stiffness of a trace element is a gross simplification, and its 662

limitation will become obvious when I analyze the solubility of noble gas elements. 663

664

Why do Ca-bearing minerals have high trace element solubility? 665

Solubility of trace elements is sensitive to minerals. Most trace elements have 666

much higher solubility in clinopyroxene than olivine (e.g., (Witt-Eickschen and O'Neill, 667

2005)). Similarly, the trace element solubility in Ca-perovskite is higher than that in Mg-668

perovskite (e.g., (Corgne et al., 2004; Hirose et al., 2004)). Common to these two cases is 669

that the solubility of trace elements is higher in a mineral that contains Ca than those that 670

do not contain Ca. 671

Here I take an example of Ca-perovskite and Mg-perovskite (bridgmanite) for 672

which a detailed study was conducted (Hirose et al., 2004). Although there is a large 673

difference in Dimineral/melt between Ca-perovskite and Mg-perovskite in the ionic size 674

versus Dimineral/melt plot (the Onuma diagram) implying that there is no large difference in 675

Dimineral/melt ,elastic between them (Fig. 1c). Therefore I conclude that most of the difference 676

between Ca-perovskite and Mg-perovskite (bridgmanite) is caused by the difference in 677

Dimineral/melt , dielectric term. The main physical property that controls Di

mineral/melt ,dielectric is 678

(static) dielectric constant, κ (equation (19)). The static dielectric constant of a mineral 679

33

depends on the polarizability of ions contained in a mineral (e.g., (Kittel, 1986)). Among 680

various cations in a typical mantle minerals, Ca has anomalously large polarizability due 681

to its large ionic size (Shannon, 1993). Consequently, a mineral that contains a large 682

amount of Ca has a large dielectric constant and hence leads to the high solubility of trace 683

elements. 684

685

Partitioning of noble gases 686

In previous sections, I pointed out that there are fundamental limitations of the 687

previous models of element partitioning in incorporating the elasticity of trace elements. 688

This problem becomes serious when one deals with noble gas elements whose elastic 689

constants are much lower than those of the host minerals (e.g., ~2-4 GPa (Devlal and 690

Gupta, 2007; Jephcoat, 1998) as compared to ~120 GPa for olivine). 691

Brooker et al. (2003) suggested that the observed trend for partitioning of noble 692

gases showing the weak dependence on noble gas atomic size (Fig. 1e) can be attributed 693

to weak effective elastic constants (a ‘zero charge’ model). However, such an explanation 694

is misleading for two reasons. First, the weak dependence of partition coefficient on the 695

atomic size of noble gas elements observed in the diagram such as Fig. 1e does not mean 696

that the solubility of noble gas elements in olivine and diopside depends weakly on the 697

atomic size of noble gas elements. The partition coefficient shown in Fig. 1e is the ratio 698

of the solubility of noble gas in a mineral to that in a melt (see equation (12), 699

Dnoble gasmineral/melt =

Cnoble gasmineral

Cnoble gasmelt ; Cnoble gas

mineral ,melt : concentration of noble gas in mineral (melt)). The 700

experimental observations shown in Fig. 1e indicate that Dnoble gasmineral/melt is weakly dependent 701

on the size of the noble gas atom. But Cnoble gasmelt is strongly dependent on the size of the 702

34

noble gas atom (Carroll and Stolper, 1993; Heber et al., 2007; Shibata et al., 1998; 703

Shibata et al., 1994) (Fig. 1f). Therefore, one must conclude that Cnoble gasmineral is strongly 704

dependent on the size of noble gas atoms. 705

To illustrate this point, I calculated Cnoble gasmineral for olivine and diopside from the 706

results shown in Fig.1e ( Dnoble gasmineral/melt ) and Fig. 1g ( Cnoble gas

melt ). Fig. 8 shows a plot of 707

partition coefficients of noble gases between olivine (or diopside) and the melt multiplied 708

by the solubility of noble gases in the melts, Dnoble gasmineral /melt ⋅Cnoble gas

melt . Essentially this is a plot 709

of the solubility (coefficient) of noble gases, Cnoble gasmineral = Dnoble gas

mineral/melt ⋅Cnoble gasmelt( ) , in olivine710

and diopside as a function of the size of noble gas atoms. This plot shows that the 711

solubility of noble gases in olivine and diopside decreases substantially with the size of 712

the noble gas atom. A similar trend was reported for the noble gas solubility in 713

bridgmanite (Shcheka and Keppler, 2012) (Fig. 1f). I conclude that the solubility of noble 714

gases in olivine, diopside and bridgmanite decreases strongly with the atomic size of 715

noble gas, and therefore these results are inconsistent with the ‘zero-charge’ model by 716

(Brooker et al., 2003). 717

Second, the model by (Brice, 1975) does not include the stiffness of the trace 718

element and the concept of “site-elasticity” in which one invokes the stiffness of the trace 719

element does not have a sound physical basis as discussed before. The elastic constant in 720

the Brice model is the elastic constant of the matrix. So even though a noble gas element 721

has ‘zero charge’, one should not make the effective elastic constant = 0 if one were to 722

use the Brice model. However, my model in its simplest form also fails to explain this 723

observation. If one uses experimentally determined bulk moduli of noble gas elements 724

35

(Devlal and Gupta, 2007; Jephcoat, 1998) with my theory (equation (20)), one would 725

predict that the effective elastic constant will be 3K12

K1+ 43Go

~0.3 GPa. This is too low to 726

explain the observations. One might try to explain this by dielectric polarization model as 727

Brooker et al. (2003) proposed. However, the dielectric polarization model predicts an 728

opposite trend (high solubility for a large size). 729

How can we interpret such a trend, i.e., the substantial reduction of the solubility 730

of noble gas elements with their atomic size? Let us use a strain energy model and 731

interpret the inferred “bulk modulus” of the noble gas based on a physical model of point 732

defects in ionic solids. Because the data are limited, I assume ro and using the solubility 733

versus atomic size ( r1 ) relation, I will estimate the effective elastic constant. The 734

observed trend (Fig. 8) suggests that the size of the site ( ro ) where a noble gas atom is 735

located in olivine and diopside must be smaller than 0.16 nm (atomic size of Ar). 736

Assuming that ro =0.072 nm in olivine, I get ~12 GPa (for clinopyroxene, assuming 737

r0 =0.1 nm, I get ~20 GPa). Similarly, Shcheka and Keppler (2012) estimated the 738

effective elastic constant in bridgmanite is ~35 GPa assuming that noble gas elements go 739

to the oxygen site ( r0 =0.14 nm). These effective elastic constants are substantially larger 740

than those estimated from the bulk moduli of the noble gases and the shear modulus of 741

the matrix using the definition of the effective elastic modulus, 3K12

K1+ 43Go

(~0.2-0.3 GPa). 742

The ECCs of the noble gas elements inferred from the experimental observations 743

of element partitioning are much higher than those calculated from the experimentally 744

determined elastic moduli of relevant elements. There is a possible physical explanation 745

for the inferred high ECC. When one inserts an atom into a crystalline site, then both 746

36

crystal and the atom deform to define the equilibrium size of the site (equation (19)). If 747

one uses the bulk modulus of the sphere, K1 (2-4 GPa), then the lattice strain will be 748

ε = %rr0

− 1 = K1K1+ 4

3Go

r1ro

− 1( ) and for a typical bulk moduli of noble gas element, the strain will749

be on the order of 1-2 % of r1ro

− 1( ) . For r1ro

− 1( )~20 %, the lattice strain would be ~0.3 %.750

Inferred high effective elastic strain implies that the atomic displacement near the 751

“defect” site (where a trace element replaces a host ion) is larger than expect from such a 752

model. This can be explained if one considers the force balance at the site where a noble 753

gas atom is inserted from a more atomistic point of view. When a neutral atom (e.g., a 754

noble gas atom) replaces a cation (e.g., Mg2+), then there will be excess 2- charge at the 755

site that will exert a repulsive force to the neighboring oxygen ions. As a consequence, 756

neighboring oxygen ions move outward (see (Spalt et al., 1973) for a case of a vacancy in 757

KBr). Consequently, the lattice strain caused by the replacement of a cation (e.g., Mg2+ ) 758

with a noble gas will be larger than what one expects from the simple elastic model. The 759

inferred large effective elastic constant for a noble gas could be due to this effect. In other 760

words, the noble gas solubility in minerals such as olivine is likely much lower than 761

expected from the low bulk moduli of the noble gases. I note that using a theoretical 762

approach Du et al. (2008) showed relatively large effective elastic moduli for the 763

dissolution of noble gases in minerals (Du et al., 2008). 764

How can one explain the large difference in the magnitude of noble gas solubility 765

between bridgmanite and other minerals (olivine and diopside)? To address this issue, let 766

us consider the processes of noble gas dissolution in more detail. Fig. 9 shows two 767

possible mechanisms of noble gas dissolution in minerals. In Fig. 9a, a noble gas atom, 768

37

Π , occupies the M-site vacancy and in Fig. 9b, it occupies the O-site vacancy. In both 769

cases, the concentration of noble gas atoms in the mineral is related to the concentration 770

of vacancies as 771

772

ΠΦΨ P,T , fO2 ,aSiO2( )⎡⎣ ⎤⎦ ∝ VΦ

Ψ P,T , fO2 ,aSiO2( )⎡⎣ ⎤⎦ ⋅ fΠ P,T( ) ⋅ KΠ P,T , fO2 ,aSiO2( ) (25)773

774

where ΠΦΨ is a noble gas atom occupying the Φ -site with an effective charge of Ψ (e.g.,775

A ′′rM (Ar at the M-site with effective two negative charge)), VΦΨ is a vacancy at the Φ -site776

with an effective charge of Ψ , fΠ is the fugacity of the noble gas Π , and KΠ is the 777

relevant equilibrium constant12. 778

The strain energy consideration discussed above was on KΠ. The equilibrium 779

constant, KΠ P,T , fO2 ,aSiO2( ) , depends on the excess energy of a mineral when vacancy is780

occupied by a noble gas. However, the difference in this term between olivine, diopside 781

and bridgmanite is not consistent with the difference in the noble gas solubility among 782

these minerals. Therefore I conclude that it is the difference in vacancy concentration, 783

VΦΨ⎡⎣ ⎤⎦ , that is responsible for the difference in the solubility of noble gases in different 784

minerals. 785

The concentration of vacancy depends strongly on minerals. In case of olivine and 786

diopside, the relevant vacancy is ′′VM whose concentration is ~10-5-10-4 under typical 787

upper mantle conditions (Nakamura and Schmalzried, 1983), whereas in bridgmanite the 788

12 To clarify the microscopic aspect, I used a point-defect notation, i.e., Kröger-Vink notation, ΠΦ

Ψ⎡⎣ ⎤⎦ rather than Cnoblegasmineral .

38

dominant vacancy is VO whose concentration is much higher although it depends on the 789

concentration of impurities such as Al3+ (e.g., (Brodhlot, 2000; Lauterbach et al., 2000; 790

Navrotsky, 1999)). This provides an explanation for the higher solubility of noble gases 791

in bridgmanite compared to olivine and diopside (e.g., (Brooker et al., 2003; Shcheka and 792

Keppler, 2012)). However, this model also implies that the solubility is highly pressure 793

dependent, ΠΦΨ⎡⎣ ⎤⎦ ∝ exp − PV*

RT( ) , where V* is the volume expansion associated with794

vacancy formation. 795

796

SOME APPLICATIONS 797

Water content in the mantle from mantle materials 798

Among the various elements, volatile elements such as H play important roles in a 799

number of geological processes and therefore estimating the water content in the mantle 800

is an important topic (e.g., (Karato, 2011; Peslier et al., 2010)). However, because of very 801

high mobility of H in olivine (and other minerals or melts; (Kohlstedt and Mackwell, 802

1998)), it is challenging to infer the distribution of H in the mantle. Evidence of 803

hydrogen-loss from olivine is frequently reported (e.g., (Demouchy et al., 2006; Peslier 804

and Luhr, 2006)). Two approaches have been conducted to overcome this difficulty. 805

One is to measure the water content of other minerals such as orthopyroxene 806

where hydrogen diffusion is more sluggish (inferred from the lack of diffusion profile in 807

opx; (Warren and Hauri, 2014)). In such a case, one might consider that the hydrogen 808

content in orthopyroxene is more “reliable” and could take it as a more faithful indicator 809

of H in the mantle. However, different water content between olivine and orthopyroxene 810

may also reflect the equilibrium partitioning that depends on the thermodynamic 811

39

conditions. A careful analysis must be made including the influence of the dependence of 812

hydrogen partition coefficient on the thermochemical conditions. The water partitioning 813

between olivine and orthopyroxene, 814

815

H[ ]oliH[ ]opx

∝ fH 2Oαoli

fH 2Oαopx

exp − EoliH +PVoli

H

RT( )exp −

EopxH +PVopx

H

RT( ) (26) 816

817

where αoli,opx is the fugacity coefficient, Eoli.opxH and Voli.opx

H are energy and volume change 818

associated with hydrogen dissolution in olivine and orthopyroxene respectively. Because 819

all of these parameters are different between olivine and orthopyroxene (Kohlstedt et al., 820

1996; Mierdel et al., 2007), the water partition coefficient, H[ ]oliH[ ]opx

, changes with the821

thermodynamic conditions by more than a factor of 10 (Dai and Karato, 2009). 822

Particularly important is the fact that in most cases, H[ ]oliH[ ]opx

∝ fH 2O1/2 , and consequently, the823

partition coefficient of water (hydrogen) between olivine an opx is depends on water 824

fugacity. Consequently, under the environment where water fugacity is low (e.g., the 825

lithosphere), the partition coefficient is low and much of water (hydrogen) in the 826

lithosphere goes to orthopyroxene. In many literatures, the observed low H[ ]oliH[ ]opx

in the827

lithosphere is interpreted to be a results of hydrogen loss from olivine, and the water 828

content in orthopyroxene is used to estimate the water content in the lithosphere 829

assuming the partition coefficient determined at high water fugacity (e.g., (Warren and 830

Hauri, 2014)). This method could lead to an over-estimate of the water content in the 831

40

lithosphere. Also, I note that (Hauri et al., 2006; Tenner et al., 2009) and (Mierdel et al., 832

2007) reported quite different depth dependence of H solubility and partitioning. 833

Ce/H ratio ( Ce[ ]H[ ] ) of basaltic magma is often used to infer the hydrogen content in834

the source region (e.g., (Dixon et al., 2002)). Again there are two concepts behind this 835

approach. First, both Ce and H are “incompatible elements” and go mostly to the melt 836

upon partial melting. The degree to which pre-existing Ce and H in the rock goes to melt 837

depends on the partition coefficients (if everything occurs as equilibrium process). The 838

assumption behind this is that this ratio is nearly constant and hence by knowing the 839

concentration of Ce, one could get some idea about the H content in the source region. 840

Also the diffusion of Ce is much slower than that of H (e.g., (Chakraborty, 2010)) so Ce 841

will faithfully reflect the Ce content of the source region while H might have escaped. 842

Another also important assumption behind this exercise is that the partition coefficient of 843

Ce and H between minerals and melts does not change with physical/chemical 844

conditions. 845

Since H and Ce have different electrostatic charges (normally H + and Ce3+ ), the 846

dissolution mechanisms of H and Ce are likely different (Fig. 10). The dissolution 847

mechanisms illustrated in Fig. 10 lead to the following relationship, 848

849

Ce[ ]H[ ] ∝ aCe2O3

1/2

fH2OfO2

−1/12 exp −P 3υMg −2υCe

2 −υMgO( )RT

⎡⎣⎢

⎤⎦⎥

(27) 850

851

where aCe2O3 is the activity of Ce2O3, fO2 is oxygen fugacity (where I assumed a relation 852

′′VM[ ] ∝ fO21/6), υMgO is the molar volume of MgO, υMg is the molar volume of Mg and υCe 853

41

is the molar volume of Ce. Among various terms in the right hand side of this equation, 854

aCe2O3 and fH 2O correspond to the composition of the material, whereas other terms 855

( fO2, expP υMgO −

3υMg−2υCe2( )

RT⎡⎣⎢

⎤⎦⎥ ) depend on the physical and chemical conditions. In particular, 856

since υMgO −3υMg−2υCe

2( ) >0, the ratio Ce[ ]H[ ] increases with pressure. The depth (therefore pressure857

and temperature) at which partial melting occurs is different among different types of 858

volcanism (e.g., mid-ocean ridge volcanism versus ocean island volcanism). Therefore 859

the ratio Ce[ ]H[ ] is likely different among the rocks from different regions. Also, the860

diffusion coefficient of Ce is much lower than that of H (Van Orman et al., 2001). 861

Therefore it is possible that Ce concentration is not in chemical equilibrium. 862

863

Are noble gases compatible or incompatible elements? 864

Noble gases are often assumed to behave like incompatible elements (e.g., 865

(Allègre et al., 1996; Marty, 2012)). However, this notion is not entirely secure because 866

either a noble gas element behaves like a compatible or incompatible element depends on 867

the solubility ratio of that element between minerals and melts, and the solubility of noble 868

gases in both minerals and melts depends strongly on pressure and temperature and 869

minerals. Consequently, it is possible that the behavior of the noble gas elements, either 870

compatible or incompatible, depends on the conditions at which melts and minerals co-871

exist. 872

There have been some challenges to the common belief of incompatible element 873

behavior of noble gases such as Ar. For instance, Watson et al. (2007) published the 874

results suggesting that Ar is a compatible element in the upper mantle although most of 875

42

previous studies show that all the noble gas elements behave like incompatible elements 876

in the upper mantle (e.g., (Broadhurst et al., 1992; Brooker et al., 2003)). Similarly, 877

Shcheka and Keppler (2012) published the experimental results showing high solubility 878

of Ar in bridgmanite suggesting that Ar might behave like a compatible element in the 879

lower mantle, although the solubility of other heavier noble gases is lower. 880

However, the validity of the conclusions by (Watson et al., 2007) is questionable. 881

The solubility of Ar reported by (Watson et al., 2007) are substantially higher than any 882

other results including those by (Hiyagon and Ozima, 1986) who reported relatively high 883

partition coefficient (high solubility) that is considered to be caused by inclusions (e.g., 884

(Broadhurst et al., 1992)). If we focus on the results where the influence of inclusions 885

was minimized (e.g., (Broadhurst et al., 1992)), the difference is even larger. The reason 886

for the reported high solubility is unknown but one possibility is that this is due to the 887

anomalous properties near the surface13 (see also (Pinilla et al., 2012)). In contrast, 888

Shcheka and Keppler (2012) measured the bulk composition and showed that 889

bridgmanite has much higher solubility of Ar than ringwoodite, and olivine. They also 890

found a systematic trend in the solubility of various noble gas elements (Fig. 1e). 891

In order to address the question of either a given noble gas behaves like an 892

incompatible element or compatible element during melting or crystallization, it is 893

necessary to compare the solubility of each noble gas element in minerals and melts. 894

I assume the results by (Broadhurst et al., 1992) on the solubility of Ne, Ar, Kr and Xe in 895

olivine and those in bridgmanite by (Shcheka and Keppler, 2012) in comparison with the 896

13 Watson et al. (2007) used near surface ~ 60 nm layers. They checked the crystallinity of studied regions by electron-back scattered pattern (EBSD) but this does not prove that these regions are defect-free.

43

experimental results on the solubility in the melts by (Heber et al., 2007). Either a noble 897

gas element behaves like a compatible or incompatible element depends on the partition 898

coefficient, D = Cmineral

Cmelt . A few assumptions are made in this analysis. First, the solubility 899

results by (Broadhurst et al., 1992) are reported as Π[ ] / fΠ (solubility divided by the900

fugacity of relevant element). However, the solubility depends on fugacity as well as the 901

free energy difference between a mineral with impurity and pure mineral as 902

Π[ ] ∝ fΠ exp − E*+PV *

RT( ) (see equation (13)). Therefore one needs to make a correction for903

the exp − E* +PV*

RT( ) term. The reported values by (Shcheka and Keppler, 2012) are directly904

Π[ ] but the results are at P=25 GPa (T=1873-2073 K). Therefore in order to discuss the905

partitioning in the whole lower mantle (P=24 to 135 GPa, T=2000-4000 K), one needs a 906

large extrapolation in the exp − E* +PV*

RT( ) term. Given a vacancy model (Fig. 9), V* is907

essentially the volume change associated with vacancy formation that is approximately 908

the volume of ion that is replaced with a noble gas14, and E* can be estimated from the 909

experimental results (Fig. 1e) using the strain energy model. 910

Ar solubility in the melts at ~25 GPa calculated from the data at ~10 GPa 911

(Chamorro-Perez et al., 1998; Schmidt and Keppler, 2002) is ~0.1 wt % in olivine melt, 912

and 0.5-0.8 wt % tholeiite melt. The Ar solubility in bridgmanite at ~25 GPa is ~0.5-1 wt 913

% (Shcheka and Keppler, 2012). This means that the frequently made assumption that Ar 914

is incompatible element (e.g., (Allègre et al., 1996; Marty, 2012)) is not valid at least in 915

the shallow lower mantle, and bridgmanite will work as a reservoir for Ar in the shallow 916

lower mantle. For other noble gas elements, data are limited, but Xe has higher solubility 917

14 This is based on the fact that oxygen is highly non-ideal gas at pressures higher than ~1 GPa.

44

in the melt than in bridgmanite, i.e., Xe is incompatible element at the shallow lower 918

mantle. 919

However, the behavior of noble gas elements under higher pressures is not 920

constrained. A theoretical model for the solubility in the melt suggests a modest decrease 921

in solubility at higher pressures (Guillot and Sarda, 2006), while a vacancy model (see 922

Fig. 9) would predict a stronger effect (with V*=5 cc/mol, increase in pressure by 50 GPa 923

will reduce the solubility by a factor of ~105) whereas the pressure effect on the solubility 924

in melt is much less according to (Guillot and Sarda, 2006). Consequently, it is expected 925

that the compatible element behavior of Ar is limited to the shallow lower mantle 926

conditions. This hypothesis needs to be tested by experiments. 927

928

SUMMARY AND CONCLUDING REMARKS 929

Extensive experimental studies on trace element partitioning have revealed 930

various trends including the importance of the difference in the size of the trace element 931

and the size of the ion that the trace element replaces. The nature of element partitioning 932

between two materials depends on how those materials accommodate “impurities”. 933

Physics and chemistry of point defects is highly relevant to understand the dissolution of 934

trace elements. A continuum model of point defects (e.g., (Eshelby, 1951, 1954, 1956; 935

Flynn, 1972)) and the basics of point defect chemistry (e.g., (Kröger and Vink, 1956)) 936

can be used to explain a majority of observations. However, I also note that some 937

atomistic details need to be incorporated in case of charged defects (e.g., (Mott and 938

Littleton, 1938)) to explain the inferred magnitude of the strain field. In melts, impurities 939

are accommodated by a more flexible structure. A hard sphere model (Barrat and Hansen, 940

45

2003; Guillot and Sarda, 2006; Guillot and Sator, 2012; Jing and Karato, 2011) provides 941

a good framework to explain various behavior of trace element solubility in the melts. 942

One important general conclusion is that the solubility and/or the partition 943

coefficient of any elements depends on minerals and melts as well as pressure, 944

temperature and other chemical parameters (such as oxygen fugacity and water fugacity). 945

Consequently, partition coefficients likely change with physical and chemical conditions. 946

Results obtained under limited conditions should not be applied to other conditions 947

without appropriate corrections. Experimental studies under a broad range of conditions 948

are important to understand the behavior of elements in Earth and planetary interiors. 949

950

ACKNOWLEDMENT 951

I thank Keith Putirka for inviting me to write this review for the centennial 952

volume of American Mineralogist. I thank Paul Asimow, Jon Blundy, Bernard Marty, 953

Eiji Ohtani, Jim Van Orman, Zhengrong Wang and Bernie Wood for discussions. Jon 954

Blundy kindly provided me with a digital data set corresponding to Fig. 5 of (Blundy and 955

Wood, 2003) that was used to prepare Fig. 7 in the present paper. 956

I thank Jon Blundy and Cin-Ty Lee for constructive reviews. 957

This work is partially supported by NSF and NASA. 958

959

46

Figure Captions 960

Fig. 1 Examples of some observations on element partitioning (solubility) 961

a. Element partition coefficient between diopside and silicate melt (Blundy and 962

Wood, 2003) (P=3 GPa, T=1930 K) 963

b. Trace element partition coefficients between (i) Mg-perovskite and silicate melt 964

and (ii) Ca-perovskite and silicate melt (Hirose et al., 2004) (P=25-27 GPa, T=2670-965

2800K) 966

c. Trace element partition coefficients between (i) Mg-perovskite and silicate melt 967

and (ii) Ca-perovskite and silicate melt plotted as a function of the size of trace element, 968

i.e., the Onuma diagram (Hirose et al., 2004) (P=25-27 GPa, T=2670-2800K)969

d. The partition coefficient of hydrogen between olivine and orthopyroxene (Dai and 970

Karato, 2009) 971

e. The partition coefficient of noble gas between olivine and silicate melt (Brooker 972

et al., 2003; Heber et al., 2007) (P=0.1 GPa, T=1530 K) 973

f. The solubility of noble gas elements in bridgmanite at P=25 GPa, T=1873-2073 K 974

(Shcheka and Keppler, 2012) 975

g. The solubility coefficient of noble gas in silicate melts at P=0.1 GPa and T~1530 976

K (Heber et al., 2007) 977

978

Fig. 2 Two ways of examining the element partitioning between a solid (a mineral) and 979

a liquid (melt) 980

(a) Direct exchange of a trace element (i) and the host ion (h) (a model used by981

(Blundy and Wood, 1994)) 982

47

(b) The same process can be envisioned as the dissolution of a trace element (i) in 983

a solid (a mineral) and a liquid (melt) (a model used in this paper). 984

985

Fig.3 A diagram showing the process of replacement of an ion with the radius ro with a 986

trace element with the radius r1 987

The final size of the site ( %r ) is between initial size ( ro ) and the size of the trace 988

element ( r1) and is determined by the size difference and the elastic properties of the 989

matrix and the trace element. 990

Fig. 4 A plot of %r

ro−1

r1ro

−1= K1

K1+ 43Go

= K1/G01+ 4

3 K1/G0( ) against K1 / G0 ( %r : the size of the site after a trace991

element occupies replaces the pre-existing cation, ro : the size of the site before a trace 992

element goes to the site (size of the cation), r1 : the size of the trace element, K1 : the bulk 993

modulus of the trace element, Go : shear modulus of the matrix) 994

If the trace element is soft ( K1 / G0 → 0; e.g., a noble gas element), then %r ≈ ro , 995

whereas if the trace element is stiff ( K1 / G0 → ∞ ), %r ≈ r1 . The assumption by (Brice, 996

1975) of %r ≈ r1 would be valid only for an infinitely stiff trace element, but not for weak 997

elements such as the noble gas elements. 998

999

Fig. 5 A schematic diagram showing the processes of trace element dissolution in a 1000

liquid (L: liquid, A: reservoir) 1001

(a) A case where a trace element (i) replaces a host ion (h) in the liquid 1002

48

This is a case when the trace element is an ion occupying a cluster in the liquid and in the 1003

reservoir. The green hexagons in these figures show the clusters each of which contains a 1004

cation and oxygen ions. 1005

(b) A case where a trace element (i) occupies the void space1006

This would be a preferred case when the trace element is neutral (e.g., noble gas). 1007

Dissolution of a trace element (noble gas atom) occurs as an addition to the liquid not as 1008

an exchange between the liquid (L) and the reservoir (A). 1009

1010

Fig. 6 Plots of normalized trace element solubility ( Cimineral ) corresponding to the elastic 1011

strain energy model against the size of the trace element corresponding to three models 1012

summarized in Table 2 ( r1 : size of the trace element, r0 : size of the site to which a trace 1013

element goes) 1014

K1 =100 GPa ( = Ko ), G0 =80 GPa, r0 = 0.1 nm, T=1600 K 1015

The comparison is made after normalizing that the strain energy at r1/ro=1 is 1016

common. Such a diagram can be directly translated to a diagram for the partition 1017

coefficient ( Dimineral/melt = Ci

mineral / Cimelt ) only when the concentration of trace element 1018

( Cim elt ) is independent of element. 1019

1020

Fig. 7 Plots showing the correlation of experimentally determined effective elastic 1021

constant EEC( )obs with the effective elastic constant from various models EEC( )calc 1022

amp: amphibole, cpx: clinopyroxene, gt: garnet, oli: olivine, opx: orthopyroxene, 1023

pla: plagioclase, woll: wollastonite (data from (Blundy and Wood, 2003)), unit of EEC is 1024

GPa 1025

49

(a) A comparison with the Brice model EEC( )calcBrice = 0.225 ⋅ Zo / ro + roxy( )4 ( Zo : electric 1026

charge of ion at the site where a trace element is dissolved, ro : ionic radius of the host 1027

ion, roxy : ionic radius of oxygen ion) 1028

A thick line corresponds to EEC( )obs = EEC( )calcBrice ( χ 2 =106). 1029

(b) A comparison with the Blundy and Wood model EEC( )calcBW = 1.12 ⋅ Z1 / ro + roxy( )31030

( Z1 : electric charge of the trace element) 1031

A thick line corresponds to EEC( )obs = EEC( )calcBW ( χ 2 =40). 1032

(c) A comparison with the present model EEC( )calcKarato = 3K1

2

K1+ 4Go3

1033

K1 is the bulk modulus of a cation-oxygen polyhedron (= 0.15 ⋅ Z1 / ri + roxy( )4 , ri : ionic 1034

radius of a trace element i). Since several different ions are used to determine EEC( )obs , I 1035

used an average value, K1 (average on various i). A thick line corresponds to 1036

EEC( )obs = EEC( )calcKarato ( χ 2 =18).1037

1038

Fig. 8 Solubility of noble gases in olivine and diopside ( Cnoble gasmineral ) calculated from the 1039

partitioning coefficient Dnoble gasmineral/melt and noble gas solubility in the melt Cnoble gas

melt using a 1040

formula Cnoble gasmineral = Dnoble gas

mineral/melt ⋅Cnoble gasmelt 1041

The data shown in Fig. 1d and Fig. 1f are used. The results correspond to P=0.1 1042

MPa and T~1550 K. Solubility of noble gases in the melt increases with pressure linearly 1043

to ~10 GPa (Guillot and Sarda, 2006). Solubility of noble gases in minerals also likely 1044

50

increases with pressure linearly in the low-pressure regime (<0.1 GPa) (Henry’s law), but1045

the pressure dependence at high pressures was not studied. 1046

1047

Fig. 9 Dissolution of a noble gas element, Π , in a mineral via a vacancy mechanism 1048

A noble gas atom ( Π ) goes into a vacant Φ -site to form a point defect ΠΦΨ ( Π 1049

occupying the Φ -site with an effective charge of Ψ ). A vacancy at the M-site ( ′′VM ) is 1050

preferred in olivine and diopside while a vacancy at the O-site ( VO) is preferred in 1051

bridgmanite. The charge compensating defects are FeM (ferric Fe at the M-site) in 1052

olivine and diopside, and ′e (free electron) in bridgmanite. 1053

1054

Fig. 10 Models of dissolution of (a) H (hydrogen) and (b) Ce in olivine 1055

(a) H2O reacts with olivine to form H-bearing olivine (as 2H( )M× + MgOsurface)1056

This model predicts H[ ] ∝ fH 2O P,T( ) ⋅exp − P⋅υMgORT( ).1057

(b) Ce2O3 reacts with olivine to form Ce-bearing olivine (as 2CeM + ′′VM + 3MgOsurface)1058

This model predicts Ce[ ] ∝ aCe2O3

1/2 ⋅ ′′VM[ ]−1/2 exp − Δυ2RT( ) ∝ aCe2O3

1/2 ⋅ fO2−1/12 exp − P 3υMg −2υCe( )

2RT( ).1059

1060

1061

51

1062 Table 1 Definition of symbols 1063 1064

Nernst partition coefficient of element i between phase Y and X

mass fraction of element i in a phase Y (X)

DiY /X = Ci

Y

CiX( ) molar partition coefficient of element i between phase Y and X

CiY ,X molar fraction of element i in a phase Y(X)

KiY /X = ai

Y

aiX( ) equilibrium constant of element i between phase Y and X

aiY ,X activity of element i in a phase Y(X)

μX chemical potential of a phase X fX fugacity of a fluid phase X ro radius of a lattice site at which a trace element is to be placed r1 radius of a trace element before placed into the crystal site

size of the crystal site after the placement of a trace element lattice strain caused by the replacement of a host ion with a trace element

K1 bulk modulus of a trace element Go shear modulus of the matrix (crystal) κ static dielectric constant of the matrix

ξ = K1K1+ 4

3Go( ) relative contribution from the trace element and the matrix to strain energy

EEC effective elastic constant of a site with a trace element χ 2 measure of the fit of a model to the data Z1 electrostatic charge of a trace element roxy radius of oxygen ion A ′′rM Ar at the M-site with effective charge of 2-

1065 1066 1067

52

Table 2 Equations for strain energy of trace element dissolution*, ** 1068

All Δu are for a defect (for per mole, one should multiply by NA Avogadro 1069

number). 1070

1071

author strain energy

Nagasawa (1966) Δuelatsic = 8πr0

3 KoGoKo+ 4

3Go

r1r0

− 1( )21+ Ko

Ko+ 43 Go

r1r0

− 1( )⎡⎣⎢

⎤⎦⎥

T

Blundy and Wood (1994) (Brice, 1975) Δuelastic = 6π r0

3 KoGo

Ko + Go3

r1r0

− 1( )21+ 2

3r1r0

− 1( )⎡⎣

⎤⎦

(T-2)

Karato (this study) Δuelastic = 6π r0

3 K12

K1+ 43G0

r1r0

− 1( )21+ K1

K1+ 43G0

r1r0

− 1( )⎡⎣⎢

⎤⎦⎥ (T-3)

1072

In equations (T-1) and (T-2), I transformed E (Young’s modulus) to a combination of 1073

bulk modulus and shear modulus using E = 9KG3K+G . 1074

ro : radius of the site into which a trace element is inserted 1075

r1: radius of the trace element 1076

K1: bulk modulus of the trace element 1077

Ko: bulk modulus of the matrix (host crystal) 1078

Go : shear modulus of the matrix (host crystal) 1079

1080

53

References 1081

Allan, N.L., Blundy, J.D., Purton, J.A., Lavrentiev, M.Y., and Wood, B.J. (2001) Trace 1082

element incorporation in minerals and melts, in: Geiger, C.A. (Ed.), EMU Notes 1083

in Mineralogy. Eötvös University Press, Budapest. 1084

Allègre, C.J. (1982) Chemical geodynamics. Tectonophysics, 81, 109-132. 1085

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1297

1298

61

Appendix 1: a modified strain energy model 1299

I consider the elastic strain energy model to calculate the energy change 1300

associated with the replacement of an ion in a crystal with another one with different size. 1301

In the elastic strain energy model, all materials involved are considered to be elastic 1302

media. Accordingly, both the matrix and the trace element are treated as elastic media. 1303

Treating a trace element as an elastic medium is a gross simplification. However, by 1304

assigning a bulk modulus to the trace element, it is possible to evaluate the influence of 1305

“stiffness” of a trace element on the strain energy. 1306

When a crystal is treated as an isotropic elastic medium, the displacement in the 1307

matrix and the spherical inclusion is given by (e.g., (Flynn, 1972)), 1308

1309

ru0,1 = A0,1

r3 + B0,1( ) rr (A-1) 1310

1311

where suffix 0 refers to those for the matrix and 1 to the trace element, and A0,1 and B0,1 1312

are constants that are to be determined by the boundary conditions. The equation (A-1) 1313

has 4 unknowns, A0,1 and B0,1 . The boundary conditions are: (1) σ rr R( ) = 0 (R is the1314

radius the crystal (homogeneous stress caused by pressure is subtracted)), (2) σrr and u1315

are continuous at the boundary between 1 and 0 ( r = %r ≡ 1+ ε( )r0 ). Note that the1316

displacement of the boundary, i.e., ε , is also an unknown that must be determined by 1317

solving the force balance and displacement continuity equations. 1318

The solution to (A-1) is somewhat tricky to obtain because of the effects of the 1319

image force, i.e., the condition σ rr R( ) = 0 (Eshelby, 1951, 1954). We consider first a1320

62

finite crystal with a finite radius R and consider the proper boundary conditions including 1321

the ones at the surface (r=R). Then we let R → ∞ . The condition of zero (excess) normal 1322

stress at r=R leads to 1323

1324

B0 = 4G03K0

A0

R3 . (A-2) 1325

1326

Note that although B0 becomes vanishingly small at R → ∞, it leads to a finite volume 1327

change of a crystal due to the effect of the image force (Eshelby, 1951, 1954). The 1328

volume change of a crystal due to this displacement is 1329

1330

Δυc = 4π R2u R( ) = 4π A0Ko + 4

3GoKo

= 12π A01−ν0( )1+ν0

. (A-3) 1331

1332

In addition, there is an explicit volume change caused by the addition of a trace element. 1333

Adding the volume change by replacing one atom (ion) with another, the net change in 1334

the volume of the whole system is given by 1335

1336

Δυ = Δυc + υ1 −υ0( ) = 12π A01−ν0( )1+ν0

+ υ1 −υ0( ) (A-4) 1337

1338

where υ1 are the volume of mineral after the trace element is dissolved and υ0 is the 1339

volume of the mineral before the trace element dissolution (the volume difference 1340

υ1 −υ0( ) may correspond to the volume change associated with the formation of point1341

defects). The normal stress at the boundary ( r = %r ≡ 1+ ε( )r0) from the inclusion comes1342

63

from the initial pressure + displacement. The conditions of continuity of stress and 1343

displacement lead to 1344

1345

− 4G0A0%r3 + 3K0B0 = 3K1 1− r1

%r( ) (A-5a) 1346

A0

%r2 + B0 %r = B1%r = %r − r0 (A-5b) 1347

1348

where K0,1 are the bulk moduli of the host crystal and the trace element respectively.1349

From (A-5b), B1%r = %r − r0 = B1r0 1+ ε( ) = r0ε , so that B1 = ε1+ε . Using (A-2) and taking1350

the limit of R → ∞ , one obtains A0 = ε 1+ ε( )2 r03.1351

Therefore the coefficients in equation (A-1) are given by, 1352

1353

A0 = ε 1+ ε( )2 r03 (A-6a) 1354

A1 = 0 (A-6b) 1355

B0 = ε 1+ ε( )2 4G03K0

r0R( )3

(A-6c) 1356

B1 = ε1+ε . (A-6d) 1357

1358

Inserting these relations into (A-5a) and ignoring the terms containing r03

R3 , one obtains 1359

1360

ε = β r1r0

−1( ) (A-7) 1361

1362

64

with β ≡ K1K1+ 4

3Go. Therefore for a very stiff trace element ( K1 >> G0 ), ε ≈ r1

r0−1 and ˜ r ≈ r11363

whereas for a very soft trace element (e.g., noble gas elements), K1 / Go 1, so ε ≈ 0 1364

and ˜ r ≈ r0 .1365

The enthalpy associated with the incorporation of trace element is given by 1366

1367

Δhela = Δuela + PΔυ (A-8) 1368

1369

where Δuela is the strain energy and Δυ is the volume change of a crystal due to the 1370

incorporation of a trace element. From (A-4) and (A-6a), the volume change is given by 1371

1372

Δυ = 4πro3 r1

r0−1( ) 1+ Ko

Ko+ 43Go

r1r0

−1( )⎡⎣⎢

⎤⎦⎥

2+ 4π

3 r03 r1

3

r03 −1⎛

⎝⎞⎠. (A-9) 1373

1374

The volume change due to this process is a fraction of atomic volume and is small 1375

compared to the volume change associated with vacancy formation. 1376

The strain energy can be calculated as 1377

1378

Δuela = 4π w1 r( )0%r∫ r2dr + lim

R→∞w0 r( )%r

R∫ r2dr⎡

⎣⎢⎤⎦⎥

(A-10)1379

where 1380

w0,1 = λ0,12

dur0,1

dr + 2 ur0,1

r( )2+ μ0,1

dur0,1

dr( )2+ 2 ur

0,1

r( )2⎡

⎣⎢

⎤

⎦⎥ (A-11)1381

1382

65

are the strain energy densities in the host crystal (0) and in the trace element (1) 1383

respectively and where λ0,1, μ0,1 are the Lamé constants of the matrix (“0”) and the trace 1384

element (“1”). 1385

From (A-5), (A-6), (A-7) and (A-11), one gets, 1386

1387

w0 = 92 K0B0

2 + 6G0A02

r6 = 2Ao2μo

4GoKo

1R6 + 3

r6⎡⎣ ⎤⎦ (A-12a) 1388

w1 = 92 K1B1

2. (A-12b) 1389

1390

Inserting equations (A-6) and with (A-10), 1391

1392

Δuela = 6π r03 ε 2 1+ ε( ) K1 + 4

3 G0( )= 6π K1

2

K1+ 43G0

r03 r1

r0−1( )2

1+ K1K1+ 4

3G0

r1r0

−1( )⎡⎣⎢

⎤⎦⎥

. (A-13) 1393

1394

The equations (A-8), (A-9) and (A-13) give the change in the elastic enthalpy, Δhela , 1395

upon the dissolution of a trace element. 1396

1397

66

Appendix 2: Electrostatic charge and effective elastic constants 1398

Strain energy model is formulated in terms of the size of the site ( ro ), the size of a 1399

trace element ( r1) and elastic constants of relevant materials (trace element and the host 1400

crystal). Comparing a theoretical relationship such as the equations (T-1) through (T-3) 1401

with the observed data on element partitioning, one can calculate the effective elastic 1402

constant. However, when one does such an exercize, the size of the site at which a peak 1403

of partition coefficient is supposedly located does not always agree with the ionic radius 1404

of the host ion (e.g., (Blundy and Dalton, 2000)). For instance, in the case of the M2 site 1405

of clinopyroxene where trace elements with 3+, 2+ and 1+ charge could go, the estimated 1406

ro from the Onuma diagram agrees well with the ionic radius of the host ion only for 1407

trace elements with 2+ charge. The inferred ro is substantially larger than the ionic radius 1408

of the host ion for trace elements with 1+ charge, and it is less than the ionic radius for 1409

trace elements with 3+ charge. 1410

This can be attributed to the influence of the charge on the atomic displacement 1411

near a point defect. When a point defect such as a vacancy is formed in an ionic crystal, it 1412

will create elastic and electric singularities. When a trace element is inserted into that site 1413

with an electric charge different from the host ion, it will generate electrostatic force to 1414

cause displacement of the ions surrounding it. For a trace element with a charge less 1415

(more) than that of the host, the force is repulsive (attractive) and the size of the site will 1416

increase (decrease). This explains the systematic shift of ro with the charge of the trace 1417

element. 1418

This effect is largest when the trace element is neutral, i.e., the noble gases. 1419

67

Appendix 3: Some notes on the estimation of EEC( )obs 1420

When the solubility of trace elements in a mineral is measured (e.g., the noble gas 1421

solubility in bridgmanite (Shcheka and Keppler, 2012)), the elastic strain energy model 1422

can be directly compared with the data on the element solubility to calculate the effective 1423

elastic constant, EEC. In most of trace elements, the available data are the partition 1424

coefficients rather than the solubility. In these cases, we need to make an assumption that 1425

the concentration of these elements in the melts is independent of the properties of the 1426

element. If this assumption is valid, then one can translate the partition coefficient as the 1427

solubility, and then compare the results with a model of element solubility (elastic strain 1428

energy model)15. 1429

There is another complication in estimating the EEC. When the EEC is calculated 1430

from the partition coefficients or the solubility, various data for a range of ionic radius (or 1431

atomic radius), r1, are used. This is not trivial because the EEC itself likely depends on 1432

the size of host ion ( ro ) and the size of the trace element ( r1 ), but the relationship 1433

between these parameters and the EEC is unknown. Furthermore, even the size of the 1434

site, ro , estimated from the Onuma diagram is sometimes different from the value 1435

expected from the ionic radius of the host ion and is treated as an unknown parameter to 1436

be determined from the experimental observations (e.g., (Blundy and Dalton, 2000)). 1437

Under these circumstances, it is justifiable to obtain a rough estimate of the EEC first 1438

assuming that it is independent of ro and r1 , and explore the correlation of the effective 1439

elastic constant with other parameters such as ro and r1 because the dependence of the 1440

EEC on these parameters is weak in comparison to the variation in the EEC. This can be 1441

15 This assumption is not valid for the noble gases.

68

seen as follows. The bulk modulus of polyhedron depends on the ionic size as 1442

Ki,o ∝ Zi,o / ri,o + roxy( )4 (corrected from (Hazen and Finger, 1979)) where roxy is the radius1443

of oxygen ion and Zi,o is the electric charge of the trace element or the host ion. When 1444

ri,o changes from 0.10 to 0.14 nm, Ki,o changes ~30% that is small compared to a 1445

variation of the EEC among different sites (a factor of ~10-100; (Blundy and Wood, 1446

2003)). Therefore such a procedure of estimating the effective elastic constant can be 1447

justified as a first-order approximation. 1448

1449

1450

Fig. 1a

cpx/melt

ionic radius (A)

part

ition

coe

ffici

ent

0.8 1.0 1.2 1.4 1.6 1.8o

Dm

iner

al/m

elt

10

10

1

10

10

2

-1

-2

Rb Ba Th U Pb Nb La Ce Sr Nd Zr Sm Ti Dy Er Yb

Ca-perovMg-perov

Fig. 1b

Ca-Pv/melt

Mg-Pv/melt

ionic radius (A)

ionic radius (A)

part

ition

coe

ffici

ent

part

ition

coe

ffici

ent

o

o

Fig. 1c

Fig. 1d

10

10

10

10

1

-4

-3

-2

-1

10-2

10-1

100

101

0.15 0.16 0.17 0.18 0.19 0.2

Ar

Kr

solu

bilit

y, w

t%

atomic radius, nm

bridgmanite

10-5

10-4

10-3

10-2

0.1 0.12 0.14 0.16 0.18 0.2

atomic radius, nm

Ar Kr Xe Ne He

min

eral

/mel

t par

titio

n co

effici

ent

olivinediopside

Fig. 1e Fig. 1f

Xe

Fig. 1g

S L

LS

(I) (II)

(a)

(b)

S

A

L

y

y

i

i

y

A

y

i

LS

i

y y

i

i

Fig. 2

2ro

2r1

2r

Fig. 3

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Fig. 4

K1 / Go

r r o1/

r 1 r o1

(a)

(b)

L L

L L

AA

A A

y y

y y

i i

ii i

Fig. 5

Cimineral

10-4

10-3

10-2

10-1

100

101

0.6 0.8 1 1.2 1.4

Nagasawa (T-1)Brice (T-2)Karato (T-3)

r1/ro

Fig. 6

102

103

104

101 102 103

ampcpxgtoliopxplawoll

Brice model

(EEC) (= , r in nm)calcBrice

(EEC

) obs

0.225 Zoro+roxy( )4

102

103

104

101 102 103

ampcpxgtoliopxplawoll

Blundy-Wood model

(EEC) (= , r in nm)calcBW

1.125 Z1ro+roxy( )3

(EEC

) obs

χ 2 = 106 χ 2 = 40

102

103

104

101 102 103

ampcpxgtoliopxplawoll

Karato model

Fig. 7c

(EEC) (= )

(EEC

)

calcKarato 3K1

2

K143Go

obs

2 18

(a) (b)

(c)

diopsideolivine

atomic radius, nm

Fig. 8

10-5

10-4

10-3

0.1 0.12 0.14 0.16 0.18 0.2

He Ne Ar Kr Xe

solu

bilit

y, w

t ppm

Fig. 9

charge-compensating defects charge-compensating defects

V

H2O MgM

MgO

2H M

Ce2O3

MgM MgM

MgMVM

MgOMgOMgOg

(a)

(b)

CeM CeM

Fig. 10